| Min(phi) over symmetries of the knot is: [-2,-2,0,1,1,2,-1,0,1,3,2,1,0,2,2,-1,0,1,-1,1,1] |
| Flat knots (up to 7 crossings) with same phi are :['6.1327'] |
| Arrow polynomial of the knot is: 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| Flat knots (up to 7 crossings) with same arrow polynomial are :['6.313', '6.623', '6.1031', '6.1201', '6.1327', '6.1378', '6.1640', '6.1697', '6.1797', '6.1833'] |
| Outer characteristic polynomial of the knot is: t^7+49t^5+101t^3+16t |
| Flat knots (up to 7 crossings) with same outer characteristic polynomial are :['6.1327'] |
| 2-strand cable arrow polynomial of the knot is: -256*K1**4*K2**2 + 448*K1**4*K2 - 416*K1**4 + 128*K1**3*K2*K3 - 64*K1**3*K3 - 704*K1**2*K2**4 + 1664*K1**2*K2**3 - 6944*K1**2*K2**2 - 128*K1**2*K2*K4 + 5288*K1**2*K2 - 2876*K1**2 + 1088*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 32*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4976*K1*K2*K3 + 96*K1*K3*K4 - 448*K2**6 + 544*K2**4*K4 - 3320*K2**4 - 528*K2**2*K3**2 - 208*K2**2*K4**2 + 2216*K2**2*K4 - 728*K2**2 + 152*K2*K3*K5 - 932*K3**2 - 302*K4**2 + 2244 |
| Flat knots (up to 6 crossings) with same 2-strand cable arrow polynomial are :['6.1327'] |
| Virtual knots (up to 6 crossings) projecting to this knot are :'vk6.17494', 'vk6.17502', 'vk6.17551', 'vk6.17559', 'vk6.24018', 'vk6.24030', 'vk6.24092', 'vk6.24099', 'vk6.36276', 'vk6.36286', 'vk6.36343', 'vk6.36351', 'vk6.43431', 'vk6.43439', 'vk6.43462', 'vk6.43468', 'vk6.55616', 'vk6.55624', 'vk6.55646', 'vk6.55653', 'vk6.60130', 'vk6.60141', 'vk6.60163', 'vk6.60172', 'vk6.65323', 'vk6.65328', 'vk6.65353', 'vk6.65363', 'vk6.68495', 'vk6.68499', 'vk6.68516', 'vk6.68524'] |
| The R3 orbit of minmal crossing diagrams contains: |
| The diagrammatic symmetry type of this knot is c. |
| The reverse -K is |
| The mirror image K* is |
| The reversed mirror image -K* is |
| The fillings (up to the first 10) associated to the algebraic genus: |
| Or click here to check the fillings |
| invariant | value |
|---|---|
| Gauss code | O1O2O3U1O4O5U2U3U5O6U4U6 |
| R3 orbit | {'O1O2O3U1O4O5U2U3U5O6U4U6'} |
| R3 orbit length | 1 |
| Gauss code of -K | O1O2O3U4U5O4U6U1U2O6O5U3 |
| Gauss code of K* | O1O2O3U4O5O4U6U1U2O6U5U3 |
| Gauss code of -K* | O1O2O3U1U4O5U2U3U5O6O4U6 |
| Diagrammatic symmetry type | c |
| Flat genus of the diagram | 3 |
| If K is checkerboard colorable | False |
| If K is almost classical | False |
| Based matrix from Gauss code | [[ 0 -2 -2 0 1 2 1],[ 2 0 1 2 2 2 0],[ 2 -1 0 1 3 2 1],[ 0 -2 -1 0 2 1 1],[-1 -2 -3 -2 0 0 1],[-2 -2 -2 -1 0 0 0],[-1 0 -1 -1 -1 0 0]] |
| Primitive based matrix | [[ 0 2 1 1 0 -2 -2],[-2 0 0 0 -1 -2 -2],[-1 0 0 1 -2 -2 -3],[-1 0 -1 0 -1 0 -1],[ 0 1 2 1 0 -2 -1],[ 2 2 2 0 2 0 1],[ 2 2 3 1 1 -1 0]] |
| If based matrix primitive | True |
| Phi of primitive based matrix | [-2,-1,-1,0,2,2,0,0,1,2,2,-1,2,2,3,1,0,1,2,1,-1] |
| Phi over symmetry | [-2,-2,0,1,1,2,-1,0,1,3,2,1,0,2,2,-1,0,1,-1,1,1] |
| Phi of -K | [-2,-2,0,1,1,2,-1,0,1,3,2,1,0,2,2,-1,0,1,-1,1,1] |
| Phi of K* | [-2,-1,-1,0,2,2,1,1,1,2,2,-1,0,2,3,-1,0,1,1,0,-1] |
| Phi of -K* | [-2,-2,0,1,1,2,-1,1,1,3,2,2,0,2,2,1,2,1,-1,0,0] |
| Symmetry type of based matrix | c |
| u-polynomial | t^2-2t |
| Normalized Jones-Krushkal polynomial | 5z^2+18z+17 |
| Enhanced Jones-Krushkal polynomial | -4w^4z^2+9w^3z^2-8w^3z+26w^2z+17w |
| Inner characteristic polynomial | t^6+35t^4+34t^2+1 |
| Outer characteristic polynomial | t^7+49t^5+101t^3+16t |
| Flat arrow polynomial | 8*K1**3 - 2*K1**2 - 4*K1*K2 - 4*K1 + K2 + 2 |
| 2-strand cable arrow polynomial | -256*K1**4*K2**2 + 448*K1**4*K2 - 416*K1**4 + 128*K1**3*K2*K3 - 64*K1**3*K3 - 704*K1**2*K2**4 + 1664*K1**2*K2**3 - 6944*K1**2*K2**2 - 128*K1**2*K2*K4 + 5288*K1**2*K2 - 2876*K1**2 + 1088*K1*K2**3*K3 + 96*K1*K2**2*K3*K4 - 896*K1*K2**2*K3 - 32*K1*K2**2*K5 - 160*K1*K2*K3*K4 + 4976*K1*K2*K3 + 96*K1*K3*K4 - 448*K2**6 + 544*K2**4*K4 - 3320*K2**4 - 528*K2**2*K3**2 - 208*K2**2*K4**2 + 2216*K2**2*K4 - 728*K2**2 + 152*K2*K3*K5 - 932*K3**2 - 302*K4**2 + 2244 |
| Genus of based matrix | 2 |
| Fillings of based matrix | [[{1, 6}, {2, 5}, {3, 4}], [{1, 6}, {2, 5}, {4}, {3}], [{1, 6}, {3, 5}, {2, 4}], [{1, 6}, {3, 5}, {4}, {2}], [{1, 6}, {4, 5}, {2, 3}], [{1, 6}, {4, 5}, {3}, {2}], [{1, 6}, {5}, {2, 4}, {3}], [{1, 6}, {5}, {3, 4}, {2}], [{1, 6}, {5}, {4}, {2, 3}], [{2, 6}, {1, 5}, {3, 4}], [{2, 6}, {1, 5}, {4}, {3}], [{2, 6}, {3, 5}, {1, 4}], [{2, 6}, {3, 5}, {4}, {1}], [{2, 6}, {4, 5}, {1, 3}], [{2, 6}, {4, 5}, {3}, {1}], [{2, 6}, {5}, {1, 4}, {3}], [{2, 6}, {5}, {3, 4}, {1}], [{2, 6}, {5}, {4}, {1, 3}], [{3, 6}, {1, 5}, {2, 4}], [{3, 6}, {1, 5}, {4}, {2}], [{3, 6}, {2, 5}, {1, 4}], [{3, 6}, {2, 5}, {4}, {1}], [{3, 6}, {4, 5}, {1, 2}], [{3, 6}, {4, 5}, {2}, {1}], [{3, 6}, {5}, {1, 4}, {2}], [{3, 6}, {5}, {2, 4}, {1}], [{3, 6}, {5}, {4}, {1, 2}], [{4, 6}, {1, 5}, {2, 3}], [{4, 6}, {1, 5}, {3}, {2}], [{4, 6}, {2, 5}, {1, 3}], [{4, 6}, {2, 5}, {3}, {1}], [{4, 6}, {3, 5}, {1, 2}], [{4, 6}, {3, 5}, {2}, {1}], [{4, 6}, {5}, {1, 3}, {2}], [{4, 6}, {5}, {2, 3}, {1}], [{4, 6}, {5}, {3}, {1, 2}], [{4, 6}, {5}, {3}, {2}, {1}], [{5, 6}, {1, 4}, {2, 3}], [{5, 6}, {1, 4}, {3}, {2}], [{5, 6}, {2, 4}, {1, 3}], [{5, 6}, {2, 4}, {3}, {1}], [{5, 6}, {3, 4}, {1, 2}], [{5, 6}, {3, 4}, {2}, {1}], [{5, 6}, {4}, {1, 3}, {2}], [{5, 6}, {4}, {2, 3}, {1}], [{5, 6}, {4}, {3}, {1, 2}], [{6}, {1, 5}, {2, 4}, {3}], [{6}, {1, 5}, {3, 4}, {2}], [{6}, {1, 5}, {4}, {2, 3}], [{6}, {2, 5}, {1, 4}, {3}], [{6}, {2, 5}, {3, 4}, {1}], [{6}, {2, 5}, {4}, {1, 3}], [{6}, {3, 5}, {1, 4}, {2}], [{6}, {3, 5}, {2, 4}, {1}], [{6}, {3, 5}, {4}, {1, 2}], [{6}, {4, 5}, {1, 3}, {2}], [{6}, {4, 5}, {2, 3}, {1}], [{6}, {4, 5}, {3}, {1, 2}], [{6}, {5}, {1, 4}, {2, 3}], [{6}, {5}, {2, 4}, {1, 3}], [{6}, {5}, {3, 4}, {1, 2}], [{6}, {5}, {4}, {1, 3}, {2}]] |
| If K is slice | False |